Correcting low-resolution measurements

ABSTRACT

Methods and systems to correct low-resolution measurements corresponding to unobservable high-resolution measurements by introducing variation in the plurality of low-resolution measurements through iteratively computing, until a termination criteria is met, perturbed values for the low-resolution measurements. The perturbed values have a higher resolution than another resolution of the low-resolution measurements. A distribution test may afterwards be performed on the perturbed values that remain after the termination criteria is met.

TECHNICAL FIELD

The present invention relates generally to a method, system, andcomputer program product for correcting measurements. More particularly,the present invention relates to a method, system, and computer programproduct for correcting and testing the normality of a plurality ofmeasurements having low resolution.

BACKGROUND

Organizations may gather and examine information from a number ofsources to obtain a complete and accurate picture of a subject.Obtaining the information may allow the organization to answer pertinentquestions, assess outcomes, conduct research and forecast futureprobability and trends.

Maintaining the integrity of research, making educated businessdecisions, and assuring product/device quality may all be bolstered byaccurate data collecting.

SUMMARY

In one aspect, a method is disclosed. The method may include receiving aplurality of low-resolution measurements, the plurality oflow-resolution measurements corresponding to a plurality of unobservablehigh-resolution measurements. Variation may be introduced in theplurality of low-resolution measurements by iteratively computing, untila termination criteria is met, corresponding perturbed values for thelow-resolution measurements. The corresponding perturbed values may havea higher resolution than another resolution of the low-resolutionmeasurements. A distribution test may then be run on final perturbedvalues that remain after said termination criteria is met.

The method may also include performing the variation introduction bycomputing, for each low-resolution measurement, a first interval thatcontains a corresponding unobservable high-resolution measurementcorresponding to said each low-resolution measurement. A randomobservation may be generated, for each low-resolution measurement, froma uniform distribution on a defined interval. Each random observationmay be transformed to be uniform on a second interval that correspondsto a distribution function such as a cumulative distribution function ofthe first interval to obtain corresponding rescaled uniformobservations. The cumulative distribution function may be based ondistribution parameters such as mean and standard deviation of saidlow-resolution measurements. Responsive to the transforming, and usingan inverse of the distribution function, said rescaled uniformobservations may be inverse transformed to obtain the correspondingperturbed values. In particular, the transforming and the inversetransforming may be repeated iteratively using new distributionparameters of the corresponding perturbed values until said terminationcriteria is met. The distribution test may be an Anderson-Darling test.The Anderson-Darling test may test for normality or for non-normality.However other tests such as other empirical distribution function (EDF)statistics tests may be used.

In another aspect, a non-transitory computer readable storage medium isdisclosed. The non-transitory computer readable storage medium storedprogram instructions which, when executed by a processor, causes theprocessor to perform a procedure that includes receiving a plurality oflow-resolution measurements, the plurality of low-resolutionmeasurements corresponding to a plurality of unobservablehigh-resolution measurements, introducing variation in the plurality oflow-resolution measurements by iteratively computing, until atermination criteria is met, corresponding perturbed values for thelow-resolution measurements, said corresponding perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements, and running, responsive to the introducing, a distributiontest on final perturbed values that remain after said terminationcriteria is met.

In yet another aspect, a computer system is disclosed. The computersystem includes at least one processor configured to perform the stepsof receiving a plurality of low-resolution measurements, the pluralityof low-resolution measurements corresponding to a plurality ofunobservable high-resolution measurements, introducing variation in theplurality of low-resolution measurements by iteratively computing, untila termination criteria is met, corresponding perturbed values for thelow-resolution measurements, said corresponding perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements, and running, responsive to the introducing, a distributiontest on final perturbed values that remain after said terminationcriteria is met.

Other technical features may be readily apparent to one skilled in theart from the following figures, descriptions, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

To easily identify the discussion of any particular element or act, themost significant digit or digits in a reference number refer to thefigure number in which that element is first introduced.

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented.

FIG. 2 depicts a block diagram of a data processing system in whichillustrative embodiments may be implemented.

FIG. 3 depicts a block diagram of an application in which illustrativeembodiments may be implemented.

FIG. 4 depicts a method in which illustrative embodiments may beimplemented.

FIG. 5 depicts a method in which illustrative embodiments may beimplemented.

FIG. 6 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 7 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 8 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 9 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 10 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 11 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 12 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 13 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 14A depicts a device from which measurements are obtained inaccordance with one or more illustrative embodiments.

FIG. 14B depicts a plot in accordance with FIG. 14A.

FIG. 14C depicts a plot in accordance with FIG. 14A.

FIG. 14D depicts a plot in accordance with FIG. 14A.

FIG. 15A depicts a device from which measurements are obtained inaccordance with one or more illustrative embodiments.

FIG. 15B depicts a plot in accordance with FIG. 15A.

FIG. 15C depicts a plot in accordance with FIG. 15A.

FIG. 15D depicts a plot in accordance with FIG. 15A.

DETAILED DESCRIPTION

The illustrative embodiments recognize that there is a need to improvethe accuracy of measurements and data in general for further examinationor research. For example, some statistical techniques may requireselecting an appropriate distribution for a plurality ofdata/measurements. The illustrative embodiments recognize that whiledistribution tests for continuous distributions may assume that sampledata are truly continuous, measurement devices may inherently have aresolution limit that may effectively round these measurements andcreate ties in the data. For example, a sample dataset that maytheoretically take any real value over a range of positive numbers, maybe obtained from measurements that are retrieved only to the nearest 10,such that the sample may have only a few distinct values, say {30, 40,50, 60, 70, 80}. While the data measured to greater precision usinghigher precision measurement devices may follow a normal distribution, atest of these relatively low-resolution data may erroneously reject ahypothesis that they follow a normal distribution. This may be observedin many practices including, for example, the estimation of processcapability statistics, demonstration that a product, such as a medicaldevice, meets a specific reliability requirement, such as 95/95confidence and reliability, and prediction of future warranty claims andthe costs associated with them. Further, a manufacturer or researchermay need to show that a set of data is compatible with a normaldistribution (or some other specific type of distribution). However, thevariation in the data may be small relative to the measurement device,e.g., it may give measurements rounded to the nearest integer or tenthof an integer. A test of normality such as the Anderson-Darling testwill tend to reject the hypothesis of normality too often if the dataare rounded too much. Presently available systems may be limited toemploying higher resolution measurement devices to repeat measurementsand worse, may not even recognize the insufficiency of the resolution ofdevices used leading to false rejections of distribution assumptionsunder a given hypothesis. Such a manner of distribution testing iserror-prone, time consuming and costly particularly if measurements haveto be repeated. Further, such a manner of distribution testing may beespecially prohibitive for applications involving sensitive data, suchas the testing of medical devices wherein false rejections may bemisleading and even damaging. The illustrative embodiments recognizethat this has been a significant and complex pain-point in the industrywhich has hitherto been unresolved with any viable systems and processeslet alone systems and processes that are applicable across manydistributions, and practical situations. Additionally, when it is notpossible to substantiate the use of a specific distribution, such as thenormal distribution, a distribution-free approach may be necessary. Suchnonparametric approaches may require much larger sample sizes, which maybe cost prohibitive, particularly when the nature of the measurement isdestructive to the part.

The illustrative embodiments described herein generally relate toadjusting for the erroneous rounding or truncation of data/measurementsby perturbing the data at each value over a relatively wider intervaland applying a defined distribution testing to the perturbed data. Bythis unconventional approach, the measurements may more closelyrepresent what a random sample from the corresponding population maylook like.

Distribution testing may be used to evaluate data distribution and totest data for normality. Many statistical tests may be parametric (i.e.,the tests may assume that the data follows a specific distribution, hasa defined shape, and can be described by a few parameters, such as amean and a standard deviation. Some data distributions include thenormal distribution, (also known as the bell curve) and distributionsthat can be transformed to a normal distribution (such as a lognormaldistribution). In addition, non-normal distributions, such as the gammaand Weibull distributions are available. For a normal distribution, mostof the data concentrations may be near the mean, or average value andthe likelihood of obtaining values away from the mean in eitherdirection may taper off the further the concentration is from the mean.Further, an Anderson-Darling statistic may be used to assess how closelydata/measurements adhere to a certain distribution. The smaller thisstatistic is for a given data set and distribution, the better thedistribution fits the data. The Anderson-Darling statistic, may forexample, be used to determine if data fits the normality assumption fora T-test. A null hypothesis (Ho) for the Anderson-Darling testhypotheses may be: The data follows a normal distribution, whereas analternative hypothesis (H1) for the Anderson-Darling test may be: Thedata does not follow a normal distribution. To determine if the datafollows the normal distribution, an appropriate p-value may be used. Ifthe p-value is less than a predetermined alpha (typically 0.05 or 0.10),the null hypothesis that the data is from a normal distribution may berejected.

However, though the benefits of the distribution testing may be limitedby the resolution of the data, presently available systems do notaddress these needs or provide adequate solutions. The illustrativeembodiments therefore recognize that by strategically reintroducingvariation into the data/measurements, false rejections of thedistribution assumptions may occur at the stated type I error rate, oralpha as described hereinafter.

An embodiment can be implemented as a software and/or hardwareapplication. The application implementing an embodiment can beconfigured as a modification of an existing system, as a separateapplication that operates in conjunction with an existing system, astandalone application, or some combination thereof.

Particularly, some illustrative embodiments provide a method thatobtains a plurality of low-resolution measurements for a test system,the plurality of low-resolution measurements corresponding to aplurality of unobservable high-resolution measurement values. The methodintroduces variation, in the plurality of low-resolution measurements byiteratively computing, until a stability criteria is met, perturbedvalues for the low-resolution measurements, said perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements. Responsive to the computing, the method runs adistribution test on the perturbed data. In the method eachlow-resolution data may have a corresponding perturbed data value.

In another embodiment, the method obtains the set of low-resolutionmeasurements by measuring the values of a property, using alow-resolution measurement device. The values may be quantitative valuesof the property and the low-resolution measurement device may round,truncate or generally imprecisely and/or inaccurately obtain measurementdata, based on, for example, a low quality of said low-resolutionmeasurement device.

This manner of correcting low-resolution measurements and distributiontesting is unavailable in the presently available methods in thetechnological field of endeavor pertaining to statistical and predictiveanalytical platforms. A method of an embodiment described herein, whenimplemented to execute on a device or data processing system, comprisessubstantial advancement of the computational functionality of thatdevice or data processing system in configuring the performance of apredictive analytic platform.

The illustrative embodiments are described with respect to certain typesof machines developing statistical and predictive analytic models basedon data records obtained from low-resolution measurements or data. Theillustrative embodiments are also described with respect to otherscenes, subjects, measurements, devices, data processing systems,environments, components, and applications only as examples. Anyspecific manifestations of these and other similar artifacts are notintended to be limiting to the invention. Any suitable manifestation ofthese and other similar artifacts can be selected within the scope ofthe illustrative embodiments.

Furthermore, the illustrative embodiments may be implemented withrespect to any type of data, data source, or access to a data sourceover a data network. Any type of data storage device may provide thedata to an embodiment of the invention, either locally at a dataprocessing system or over a data network, within the scope of theinvention. Where an embodiment is described using a mobile device, anytype of data storage device suitable for use with the mobile device mayprovide the data to such embodiment, either locally at the mobile deviceor over a data network, within the scope of the illustrativeembodiments.

The illustrative embodiments are described using specific surveys, code,hardware, algorithms, designs, architectures, protocols, layouts,schematics, and tools only as examples and are not limiting to theillustrative embodiments. Furthermore, the illustrative embodiments aredescribed in some instances using particular software, tools, and dataprocessing environments only as an example for the clarity of thedescription. The illustrative embodiments may be used in conjunctionwith other comparable or similarly purposed structures, systems,applications, or architectures. For example, other comparable devices,structures, systems, applications, or architectures therefor, may beused in conjunction with such embodiment of the invention within thescope of the invention. An illustrative embodiment may be implemented inhardware, software, or a combination thereof.

The examples in this disclosure are used only for the clarity of thedescription and are not limiting to the illustrative embodiments.Additional data, operations, actions, tasks, activities, andmanipulations will be conceivable from this disclosure and the same arecontemplated within the scope of the illustrative embodiments.

Any advantages listed herein are only examples and are not intended tobe limiting to the illustrative embodiments. Additional or differentadvantages may be realized by specific illustrative embodiments.Furthermore, a particular illustrative embodiment may have some, all, ornone of the advantages listed above.

With reference to the figures and in particular with reference to FIG. 1and FIG. 2 , these figures are example diagrams of data processingenvironments in which illustrative embodiments may be implemented. FIG.1 and FIG. 2 are only examples and are not intended to assert or implyany limitation with regard to the environments in which differentembodiments may be implemented. A particular implementation may makemany modifications to the depicted environments based on the followingdescription.

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented. Data processingenvironment 100 is a network of computers in which the illustrativeembodiments may be implemented. Data processing environment 100 includesnetwork 102. Network 102 is the medium used to provide communicationslinks between various devices and computers connected together withindata processing environment 100. Network 102 may include connections,such as wire, wireless communication links, or fiber optic cables.

Clients or servers are only example roles of certain data processingsystems connected to network 102 and are not intended to exclude otherconfigurations or roles for these data processing systems. Server 104and server 106 couple to network 102 along with storage unit 108.Software applications may execute on any computer in data processingenvironment 100. Client 110, client 112, client 114 are also coupled tonetwork 102. A data processing system, such as server 104 or server 106,or clients (client 110, client 112, client 114) may contain data and mayhave software applications or software tools executing thereon. Server104 may include one or more GPUs (graphics processing units) fortraining one or more models.

Only as an example, and without implying any limitation to sucharchitecture, FIG. 1 depicts certain components that are usable in anexample implementation of an embodiment. For example, servers andclients are only examples and not to imply a limitation to aclient-server architecture. As another example, an embodiment can bedistributed across several data processing systems and a data network asshown, whereas another embodiment can be implemented on a single dataprocessing system within the scope of the illustrative embodiments. Dataprocessing systems (server 104, server 106, client 110, client 112,client 114) also represent example nodes in a cluster, partitions, andother configurations suitable for implementing an embodiment.

Device 120 is an example of a device described herein. For example,device 120 can take the form of a smartphone, a tablet computer, alaptop computer, client 110 in a stationary or a portable form, awearable computing device, or any other suitable device. Any softwareapplication described as executing in another data processing system inFIG. 1 can be configured to execute in device 120 in a similar manner.Any data or information stored or produced in another data processingsystem in FIG. 1 can be configured to be stored or produced in device120 in a similar manner.

Test engine 126 may execute as part of client application 122, serverapplication 116 or on any data processing system herein. Test engine 126may also execute as a cloud service communicatively coupled to systemservices, hardware resources, or software elements described herein.Database 118 of storage unit 108 stores one or more measurements or datain repositories for computations herein.

Server application 116 implements an embodiment described herein. Serverapplication 116 can use data from storage unit 108 for low-resolutiondata correction and testing. Server application 116 can also obtain datafrom any client for correction and testing. Server application 116 canalso execute in any of data processing systems (server 104 or server106, client 110, client 112, client 114), such as client application 122in client 110 and need not execute in the same system as server 104.

Server 104, server 106, storage unit 108, client 110, client 112, client114, device 120 may couple to network 102 using wired connections,wireless communication protocols, or other suitable data connectivity.Client 110, client 112 and client 114 may be, for example, personalcomputers or network computers.

In the depicted example, server 104 may provide data, such as bootfiles, operating system images, and applications to client 110, client112, and client 114. Client 110, client 112 and client 114 may beclients to server 104 in this example. Client 110, client 112 and client114 or some combination thereof, may include their own data, boot files,operating system images, and applications. Data processing environment100 may include additional servers, clients, and other devices that arenot shown. Server 104 includes a server application 116 that may beconfigured to implement one or more of the functions described hereinfor low-resolution measurement correction in accordance with one or moreembodiments.

Server 106 may include a search engine configured to search measurementsor databases in response to a query with respect to various embodiments.The data processing environment 100 may also include a dedicatedmeasurement system 124 which comprises a test engine 126. The dedicatedmeasurement system 124 may be used for performing measurements ofdefined properties, via special purpose measurement devices 128 such asmedical devices, vision and imaging devices, detectors, transducers,sensors instruments used in measuring physical quantities and attributesof real-world objects and events. The dedicated measurement system 124may also be used to test samples using the test engine 126. Themeasurement system 124 may make decisions about the distributionsmeasurements belong to by performing distribution testing tomeasurements responsive to performing perturbations on low-resolutionmeasurements. For example, it may apply an Anderson-Darling test to themeasurements modified by perturbation techniques described herein, whichmay result in data having ideal statistical properties.

An operator of the measurement system 124 can include individuals,computer applications, and electronic devices. The operators may employthe test engine 126 of the measurement system 124 to make predictions ordecisions. An operator may desire that the test engine 126 performmethods to satisfy a predetermined evaluation criteria. Thus, a new andunique way to perturb data to address rounding and similar measurementissues that is effective, statistically appropriate and much moreaccurate than using the Anderson-Darling statistic and p-values on theraw, unadjusted data may be provided.

The data processing environment 100 may also be the Internet. Network102 may represent a collection of networks and gateways that use theTransmission Control Protocol/Internet Protocol (TCP/IP) and otherprotocols to communicate with one another. At the heart of the Internetis a backbone of data communication links between major nodes or hostcomputers, including thousands of commercial, governmental, educational,and other computer systems that route data and messages. Of course, dataprocessing environment 100 also may be implemented as a number ofdifferent types of networks, such as for example, an intranet, a localarea network (LAN), or a wide area network (WAN). FIG. 1 is intended asan example, and not as an architectural limitation for the differentillustrative embodiments.

Among other uses, data processing environment 100 may be used forimplementing a client-server environment in which the illustrativeembodiments may be implemented. A client-server environment enablessoftware applications and data to be distributed across a network suchthat an application functions by using the interactivity between aclient data processing system and a server data processing system. Dataprocessing environment 100 may also employ a service-orientedarchitecture where interoperable software components distributed acrossa network may be packaged together as coherent business applications.Data processing environment 100 may also take the form of a cloud andemploy a cloud computing model of service delivery for enablingconvenient, on-demand network access to a shared pool of configurablecomputing resources (e.g., networks, network bandwidth, servers,processing, memory, storage, applications, virtual machines, andservices) that can be rapidly provisioned and released with minimalmanagement effort or interaction with a provider of the service.

With reference to FIG. 2 , this figure depicts a block diagram of a dataprocessing system in which illustrative embodiments may be implemented.Data processing system 200 is an example of a computer, such as server104, server 106, or client 110, client 112, client 114, measurementsystem 124 in FIG. 1 , or another type of device in which computerusable program code or instructions implementing the processes may belocated for the illustrative embodiments.

Data processing system 200 is also representative of a data processingsystem or a configuration therein, such as device 120 in FIG. 1 in whichcomputer usable program code or instructions implementing the processesof the illustrative embodiments may be located. Data processing system200 is described as a computer only as an example, without being limitedthereto. Implementations in the form of other devices, such as device120 in FIG. 1 , may modify data processing system 200, such as by addinga touch interface, and even eliminate certain depicted components fromdata processing system 200 without departing from the generaldescription of the operations and functions of data processing system200 described herein.

In the depicted example, data processing system 200 employs a hubarchitecture including North Bridge and memory controller hub (NB/MCH)202 and South Bridge and input/output (I/O) controller hub (SB/ICH) 204.Processing unit 206, main memory 208, and graphics processor 210 arecoupled to North Bridge and memory controller hub (NB/MCH) 202.Processing unit 206 may contain one or more processors and may beimplemented using one or more heterogeneous processor systems.Processing unit 206 may be a multi-core processor. Graphics processor210 may be coupled to North Bridge and memory controller hub (NB/MCH)202 through an accelerated graphics port (AGP) in certainimplementations.

In the depicted example, local area network (LAN) adapter 212 is coupledto South Bridge and input/output (I/O) controller hub (SB/ICH) 204.Audio adapter 216, keyboard and mouse adapter 220, modem 222, read onlymemory (ROM) 224, universal serial bus (USB) and other ports 232, andPCI/PCIe devices 234 are coupled to South Bridge and input/output (I/O)controller hub (SB/ICH) 204 through bus 218. Hard disk drive (HDD) orsolid-state drive (SSD) 226 a and CD-ROM 230 are coupled to South Bridgeand input/output (I/O) controller hub (SB/ICH) 204 through bus 228.PCI/PCIe devices 234 may include, for example, Ethernet adapters, add-incards, and PC cards for notebook computers. PCI uses a card buscontroller, while PCIe does not. Read only memory (ROM) 224 may be, forexample, a flash binary input/output system (BIOS). Hard disk drive(HDD) or solid-state drive (SSD) 226 a and CD-ROM 230 may use, forexample, an integrated drive electronics (IDE), serial advancedtechnology attachment (SATA) interface, or variants such asexternal-SATA (eSATA) and micro-SATA (mSATA). A super I/O (SIO) device236 may be coupled to South Bridge and input/output (I/O) controller hub(SB/ICH) 204 through bus 218.

Memories, such as main memory 208, read only memory (ROM) 224, or flashmemory (not shown), are some examples of computer usable storagedevices. Hard disk drive (HDD) or solid-state drive (SSD) 226 a, CD-ROM230, and other similarly usable devices are some examples of computerusable storage devices including a computer usable storage medium.

An operating system runs on processing unit 206. The operating systemcoordinates and provides control of various components within dataprocessing system 200 in FIG. 2 . The operating system may be acommercially available operating system for any type of computingplatform, including but not limited to server systems, personalcomputers, and mobile devices. An object oriented or other type ofprogramming system may operate in conjunction with the operating systemand provide calls to the operating system from programs or applicationsexecuting on data processing system 200.

Instructions for the operating system, the object-oriented programmingsystem, and applications or programs, such as server application 116 andclient application 122 in FIG. 1 , are located on storage devices, suchas in the form of codes 226 b on Hard disk drive (HDD) or solid-statedrive (SSD) 226 a, and may be loaded into at least one of one or morememories, such as main memory 208, for execution by processing unit 206.The processes of the illustrative embodiments may be performed byprocessing unit 206 using computer implemented instructions, which maybe located in a memory, such as, for example, main memory 208, read onlymemory (ROM) 224, or in one or more peripheral devices.

Furthermore, in one case, code 226 b may be downloaded over network 214a from remote system 214 b, where similar code 214 c is stored on astorage device 214 d in another case, code 226 b may be downloaded overnetwork 214 a to remote system 214 b, where downloaded code 214 c isstored on a storage device 214 d.

The hardware in FIG. 1 and FIG. 2 may vary depending on theimplementation. Other internal hardware or peripheral devices, such asflash memory, equivalent non-volatile memory, or optical disk drives andthe like, may be used in addition to or in place of the hardwaredepicted in FIG. 1 and FIG. 2 . In addition, the processes of theillustrative embodiments may be applied to a multiprocessor dataprocessing system.

In some illustrative examples, data processing system 200 may be apersonal digital assistant (PDA), which is generally configured withflash memory to provide non-volatile memory for storing operating systemfiles and/or user-generated data. A bus system may comprise one or morebuses, such as a system bus, an I/O bus, and a PCI bus. Of course, thebus system may be implemented using any type of communications fabric orarchitecture that provides for a transfer of data between differentcomponents or devices attached to the fabric or architecture.

A communications unit may include one or more devices used to transmitand receive data, such as a modem or a network adapter. A memory may be,for example, main memory 208 or a cache, such as the cache found inNorth Bridge and memory controller hub (NB/MCH) 202. A processing unitmay include one or more processors or CPUs.

The depicted examples in FIG. 1 and FIG. 2 and above-described examplesare not meant to imply architectural limitations. For example, dataprocessing system 200 also may be a tablet computer, laptop computer, ortelephone device in addition to taking the form of a mobile or wearabledevice.

Where a computer or data processing system is described as a virtualmachine, a virtual device, or a virtual component, the virtual machine,virtual device, or the virtual component operates in the manner of dataprocessing system 200 using virtualized manifestation of some or allcomponents depicted in data processing system 200. For example, in avirtual machine, virtual device, or virtual component, processing unit206 is manifested as a virtualized instance of all or some number ofhardware processing units 206 available in a host data processingsystem, main memory 208 is manifested as a virtualized instance of allor some portion of main memory 208 that may be available in the hostdata processing system, and Hard disk drive (HDD) or solid-state drive(SSD) 226 a is manifested as a virtualized instance of all or someportion of Hard disk drive (HDD) or solid-state drive (SSD) 226 a thatmay be available in the host data processing system. The host dataprocessing system in such cases is represented by data processing system200.

With reference to FIG. 3 , this figure depicts a block diagram of anexample configuration for correcting and testing low-resolutionmeasurements. The example embodiment includes application 302. In aparticular embodiment, application 302 is an example of clientapplication 122 or server application 116 of FIG. 1 .

Application 302 receives a set or plurality of low-resolutionmeasurements 306 for a test system. In a particular embodiment, thelow-resolution measurements 306 represents quantitative measurementsobtained by an operator using one or more measurement devices 128. Forexample, the measurements/data may be obtained from manufacturer testingsuch as ISO (International Organization for Standardization) testing ofballoon rated burst pressures, which may enable catheter manufacturersdetermine a rated burst pressure (RBP)—the pressure at which 99.9% ofballoons can survive with 95% confidence. Further, a pin gage is a steelpin used to quickly measure the diameter of a drilled hole in metal orother material. Pin gages come in sets containing various sized pins.When measuring hole size, the diameter of the largest pin that will fitis recorded as the diameter of the hole. A pin gage measuring system mayhave poor resolution because of the relatively large differences in pingage diameter from one size gage to the next. Even further, food,beverage, pharmaceutical and medical device manufacturers may have tocarefully seal their product packaging to strict specifications so theproduct remains safe for consumption. If the seal is too weak, thepackaging may open during shipment. If the seal is too strong, aconsumer may have difficulty opening the packaging. Seal strength is themaximum force needed to separate the two layers of a seal underparticular conditions. Seal strength may be rounded to the nearestNewton per square millimeters, causing low resolution in themeasurements which may make it difficult to assess the true processcapability. In another example, air quality meters, designed to measureair velocity, pressure, gases, temperature, humidity, dust etc. may beused may be used to obtain measurements which may be of low-resolution.Of course, these examples are not meant to be limiting as measurementsfrom any continuous distribution may be included.

In the embodiment, interval determination component 304 may beconfigured to determine, based on a resolution of the low-resolutionmeasurements, a first interval known to contain an unobservablehigh-resolution measurement value that corresponds to a low-resolutionmeasurement value. This may be performed for all low-resolutionmeasurement values in a data set. Random observation generationcomponent 308 may generate, for each low-resolution measurement value,random observations from a uniform distribution on an interval (0,1).Data perturbation component 310 may transform, using the transformationcomponent 312, the random observations to be uniform on a secondinterval that is based on a cumulative distribution function of thenormal distribution (or of another distribution being tested) to obtainrescaled uniform observations. The rescaled uniform observations may betransformed back using the inverse cumulative distribution function toobtain perturbed values. This may be repeated under new statistics untila termination criteria is achieved as described hereinafter. Further,the distribution test component 314 may perform a test of whether theperturbed values follow a predefined distribution responsive toobtaining final perturbed values.

FIG. 4 illustrates a process 400 in which illustrative embodiments maybe implemented. The process begins in step 402, wherein process 400obtains a plurality of low-resolution measurements for a test system,the plurality of low-resolution measurements corresponding to aplurality of unobservable high-resolution measurement values. In step404, process 400 introduces variation, in the plurality oflow-resolution measurements by iteratively computing, until a stabilitycriteria is met, perturbed values for the low-resolution measurements,the perturbed values having a higher resolution than another resolutionof the low-resolution measurements. Responsive to final perturbed valuesbeing computed, the process may perform a distribution test on the finalperturbed data.

FIG. 5 illustrates a specific example process 500 the process 400 ofFIG. 4 . The process 500 may begin at step 502, wherein process 500receives a plurality of low-resolution measurements, the plurality oflow-resolution measurements corresponding to a plurality of unobservablehigh-resolution measurements (

, i=1, . . . , n). In an example, “n” number of low-resolutionmeasurements X_(i), i=1, . . . , n may be received. In step 504, process500 computes, for each low-resolution measurement, a first interval[L_(i), H_(i)] that contains a corresponding unobservablehigh-resolution measurement corresponding to said each low-resolutionmeasurement value. The first interval may be based on the range ofpossible values of the unobservable high-resolution data that would havebeen rounded to each observed low-resolution value. E.g. if alow-resolution value 13 is observed, the interval may be 12.5 to 13.5.For a situation where the low-resolution measurements are a roundedversion of the unobservable high-resolution measurement, i.e. where

X i = Δ ⁢ Round ( Δ ) , i = 1 , … , n

the first interval [L_(i), H_(i)] may be obtained as shown:

$\left\lbrack {L_{i},H_{i}} \right\rbrack = \left\lbrack {{X - \frac{\Delta}{2}},{X + \frac{\Delta}{2}}} \right\rbrack$

In step 506, process 500 generates, for each low-resolution measurement,a random observation from a uniform distribution on a defined interval(0,1). Thus, step 506 may generate “n” random observations U_(i), i=1, .. . , n. In step 508, process 500 may estimate the distributionparameters (e.g., mean and standard deviation for the normaldistribution and thus sample mean ({circumflex over (μ)}) and samplestandard deviation ({circumflex over (σ)}) for initial estimates) of thelow-resolution measurements. Said sample mean ({circumflex over (μ)})and sample standard deviation ({circumflex over (σ)}) may be estimatedas follows:

$\hat{\mu} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}X_{i}}}$$\hat{\sigma} = \sqrt{\frac{\sum_{i = 1}^{n}\left( {X_{i} - \hat{\mu}} \right)^{2}}{n - 1}}$

In step 510, the process 500 may transform each random observation U_(i)to be uniform on a second interval [F(L_(i)), F(H_(i))], to obtainrescaled uniform observations W_(i), with F being the cumulativedistribution function for the normal distribution with the estimateddistribution parameters (estimated sample mean and estimated samplestandard deviation).

The resealed uniform observations W_(i) may be computed as follows:

W _(i)=({circumflex over (F)}(L _(i))+({circumflex over (F)}(H_(i))−{circumflex over (F)}(L _(i)))U _(i)

For a normal distribution, the estimated cumulative distributionfunction may be estimated as follows, with ϕ denoting the cumulativedistribution function of the standard normal distribution:

${\overset{\hat{}}{F}(x)} = {\Phi\left( \frac{x - \overset{\hat{}}{\mu}}{\overset{\hat{}}{\sigma}} \right)}$

In step 512, process 500 obtains perturbed values (

, i=1, . . . , n) by inverse transforming the resealed uniformobservations W_(i), responsive to the transforming step of step 510 andusing an inverse of the cumulative distribution function.

{circumflex over (X)} _(i) ={circumflex over (F)} ⁻¹(W _(i)).

In step 514, process 500 may estimate distribution parameters (e.g., themean and standard deviation in the case of a normal distribution) of theperturbed values. In step 516, process 500 may determine if anevaluation/termination criteria condition is met. The terminationcriteria condition may be whether the standard deviation is stable.Responsive to determining that the termination criteria condition is notmet, process 500 obtains the estimates of step 514 for use, in step518). In other words, updated estimates of the parameters may beobtained based on

. For the normal distribution, these may be the sample mean and samplestandard deviation of these values:

$\hat{\mu} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}{\hat{X}}_{i}}}$$\hat{\sigma} = \sqrt{\frac{\sum_{i = 1}^{n}\left( {{\hat{X}}_{i} - \hat{\mu}} \right)^{2}}{n - 1}}$

In an example, the termination criteria condition is met when apercentage change of the scale parameter/standard deviation is less than0.01%) or until a predetermined maximum number of iterations (e.g., 5)is completed. Further a combination of termination criteria may be used.For example, an updated standard deviation estimate {circumflex over(ϕ)}_(new) may be compared to a previous estimate {circumflex over(ϕ)}_(old) and a termination rule

$\frac{❘{\sigma_{new} - \sigma_{old}}❘}{\sigma_{old}} \leq {{0.0}001}$

is used along with performing a maximum of 5 iterations.

Thus, process 500 repeats from step 510 using the new sample mean andthe new sample standard deviation until the termination criteriacondition is met. Upon meeting the termination criteria condition, atest such as the Anderson-Darling normality test may be performed on thefinal perturbed data and process 500 ends thereafter. Of course, theseexamples are not meant to be limiting as variations thereof may beachieved from descriptions herein.

FIG. 6 illustrates example measurements generated to simulate actualmeasurements seen from a medical device. The measurements/data form arandom sample from a normally distributed population. While this data isknown to come from a normally-distributed population, the nullhypothesis that the data is from a normally-distributed population isrejected (P-value 602 of less than 0.005, with 0.05 being a thresholdpass condition). A corresponding Anderson-Darling statistic 604 thatresults (1.258) is high. This failure result for a goodness-of-fit testfor a continuous distribution may consistently occur for measurementssampled from that continuous distribution when the measurements arerounded because of poor measurement resolution.

FIG. 7 illustrates a plot with the same measurements from FIG. 6 beingperturbed according to methods described herein. The null hypothesisthat the measurements are from a normally-distributed population passes(P-value 602 of 0.624, with 0.05 being the threshold pass condition). Acorresponding Anderson-Darling statistic 604 of the perturbed data iscomputed to be low (0.281).

Further, simulation studies were performed that may demonstrate problemsassociated with applying the Anderson-Darling test directly tolow-resolution data and may further demonstrate the superior statisticalproperties attained by methods described herein. The simulations wereperformed based on sets of 2000 columns of generated data. In somecases, samples of data from the normal distribution are simulated. Theseare cases where the Anderson-Darling test should indicate that thenormal distribution fits the data well in the vast majority of cases.Rounding of the data were carried out and varied to achieve differentratios of the rounding to the standard deviation of the data. In othercases, the Chi square methods were used to simulate non-normal data.These are cases where the Anderson-Darling test frequently is able toshow a lack of fit of the normal distribution to the data. The resultswere then used to evaluate how the distribution of p-values for therounded data with and without perturbation steps described hereincompared to those obtained when the data are not rounded. For roundeddata without our method, the distribution of p-values tends to be toolow, which results in rejecting the normal distribution with too high aprobability under the condition where the original data are normallydistributed before rounding. This worsens as the rounding ratioincreases. On the other hand, applying perturbation steps describedherein results in the rounded data stabilizing the distribution of thep-values over a range of practical rounding ratios when the originaldata are from a normal distribution.

More specifically, FIG. 8 shows an example wherein the measurementvalues are generated to follow a normal distribution. The figureillustrates tests for rounded values without perturbation-50 samples802, rounded values without perturbation-100 samples 804, rounded valueswith perturbation-50 samples 806, and rounded values withperturbation-100 samples 808 having defined rounding ratios 810(roundingwidth/standard deviation). For each defined ratio, the test is repeateda number of times(e.g., 2000 times) to find the percentage 812 of timesthe test incorrectly rejects the null hypothesis. As can be seen,introducing perturbation provides the rounded values withperturbation-50 samples 806 and the rounded values with perturbation-100samples 808 with significantly less percentage 812 of times the testincorrectly rejects the null hypothesis (y-axis, i.e. proportion oftimes the test made a mistake) than the percentage of times the testincorrectly rejects the null hypothesis for rounded values withoutperturbation-50 samples 802 and rounded values without perturbation-100samples 804. In other words, the more rounding, the worse the valueswithout perturbation performs(approaches 100% with increased rounding),whereas the values with perturbation stays steady throughout therounding.

FIG. 9 illustrates the power (ability to show that the data/measurementis not normal when it truly comes from a non-normal population) of theAnderson-Darling test in cases where the data is not generated to followa normal distribution. In this case, the higher the percentage 812 oftimes the null hypothesis that the data follows a normal distribution isrejected, the better. A chi-square distribution with different defineddegrees of freedom 902 and sample sizes 814 used. As can be seen, theperturbation provides sufficient “power” over the range of practicalrounding ratios 810 considered, and power comparable to that ofunrounded highest resolution data initially though it may graduallydecline as the rounding ratio increases. This may be expected with theloss of information imposed by rounding.

FIG. 10 illustrates that the power of the Anderson-Darling test for aT-distribution with 4 degrees of freedom and sample size of 100 may alsohave a power that follows a similar trend as shown in the chi-squareexamples of FIG. 9 . Thus, from these examples, it can be seen that themethods herein may provide reduced rejection when the data follows anormal distribution and increased rejection when the data/measurementsdo not follow a normal distribution.

FIG. 11 illustrates how well methods described herein perform or agree(percent agreement 1110) with the performance using full unroundedhigh-resolution data. Rounded values with perturbation-50 samples 1106and rounded values with perturbation-100 samples 1108 provideperformance comparable to having the unrounded high-resolution datawhereas rounded values without perturbation-50 samples 1102 and roundedvalues without perturbation-100 samples 1104 provide poor performancewith increasing rounding. The data were generated to follow a normaldistribution. More specifically, methods described herein may be basedon the idea of reintroducing some of the variation removed from the databy rounding. This may be done by means of randomly generated data. Thus,the result may also be random and may depend on the particular randomdata generated. Therefore, if the method were run again, the result maybe different—possibly very similar, but not necessarily. The methodcarefully achieves this in a way that may preserve the distribution ofthe results one would get if the data were not rounded. This may beperformed in a way that preserves as much of the specific information inthe original data as is available. However, as the rounding ratioincreases, the amount of information available about the complete datamay diminish, and the variability of the results may increase. FIG. 11thus shows how the p-values may relate to the complete-data p-values. Inparticular, FIG. 11 shows how often method discussed herein agreed withthe original-data results concerning whether or not to reject the nullhypothesis or normality.

FIG. 12 shows another example of the percent agreement 1110 wherein thedata follows a non-normal distribution. Both sets of data were generatedbased on a chi-square distribution, but the degrees of freedom arevaried to keep the power roughly the same as the number samples arevaried. For 50 rows, 6 degrees of freedom are used and for 100 rows used12 degrees of freedom are used. It can be seen that tests using roundedvalues with perturbation-50 samples-6 degrees of freedom 1206 androunded values with perturbation-100 samples-12 degrees of freedom 1208performed better than tests using rounded values without perturbation-50samples-6 degrees of freedom 1202 and rounded values withoutperturbation-100 samples-12 degrees of freedom 1204. Moreover, percentagreement 1110 for 1206 and 1208 were still high at about 80% near 0.8rounding and at about 70% with the highest rounding.

FIG. 13 shows another comparison similar to that of FIG. 12 based on a Tdistribution with 4 degrees of freedom with 100 samples each. The figureshows tests using rounded values without perturbation-100 samples-4degrees of freedom 1302 and rounded values with perturbation-100samples-4 degrees of freedom 1304. For even moderately large roundingratios, the 1302 will always reject, and hence all of the agreement isfor sets where the original data also led to rejection of the nullhypothesis. On the other hand, perturbation, as shown by 1304 continuesto be more discriminating, and has better agreement with thefull-resolution data even at the highest rounding ratio considered.

Of course, the examples of FIG. 8 -FIG. 13 are examples and are notmeant to be limiting for the methods described herein as other examplesmay be obtained in light of the descriptions herein.

In more use cases, a bend test for bone plates 1402 of FIG. 14A is shownby FIG. 14B-FIG. 14D. Bone plates are thin metal implants used to holdbone segments in the correct position while they heal after a break orother condition. A bone plate may be attached with screws to align andstabilize a broken bone. A variety of laboratory tests may be performedon bone plates to ensure their safety and efficacy. One such test, astatic bend test, may apply increasing force to the bone plate until itbreaks. Requirements may be set on the minimum force for which 99% ofthe population is expected to survive with 95% confidence. Beforedetermining whether the reliability requirements are met, it may benecessary to determine the underlying distribution of the forcemeasurements. It is common to assume that the forces required to breakthese bone plates follow a normal distribution. If the hypothesis testfor normality concludes that the data is non-normal, the user mayinstead use a non-normal distribution or a distribution-free approach todemonstrate reliability. The resulting force measurements 1404, inNewtons, may be rounded because, for example, the testing machine mayonly apply specific levels of force. For a sample size of N=28, thehistogram of FIG. 14B may indicate that the force measurements 1404 arereasonably symmetric and there isn't much evidence to suggest that theyare non-normal. However, as shown in FIG. 14C, the p-value 602 for theAnderson-Darling Normality test (P=0.013) may reject the null hypothesisof normality and lead to the conclusion that these data are from anon-normal population (with 0.05 being the rejection threshold). Furtherthe Anderson-Darling statistic 604 may be inflated, casting theconclusion of non-normality into doubt. By perturbing the forcemeasurements 1404 into perturbed measurements 1406 a p-value 602 of0.310 is obtained which may indicate that the null hypothesis ofnormality will no longer be rejected, and a normal distribution may beused when computing the necessary reliability estimates.

In another use case, the seal strength for the packaging 1504 of urinarycatheters 1502 may be highly regulated due to the risks associated withimproper packaging. Catheters may be sterilized when packaged so theymay be immediately used upon opening. The seal strength of the catheterpackaging may be tested to ensure that the device remains sterile. Anysection of the seal that is weak or compromised may provide anopportunity for entry of potential contaminants. Seal strength is theforce required to remove the seal from the packaging. Force measurements1506, in pound (force), as shown in FIG. 15B are typically rounded,making it difficult to determine the distribution of the underlyingpopulation. The population distribution may be critical for reportingstatistics such as “Cpk” that determine the capability of themanufacturing process to meet desired specifications for seal strength.As shown in the histogram of FIG. 15B, there may be no evidence tosuggest that these force measurements 1506 are from a non-normalpopulation. However, the p-value 602 for the Anderson-Darling statistic(P-Value=0.019) suggests that the null hypothesis of normality should berejected and concludes that the force measurement 1506 are non-normal.Because the data are rounded, however, the Anderson-Darling statistic604 is inflated(0.919). This may make the measurements appear to be froma significantly non-normal population when they are not. By perturbingthe measurements to correct the low resolution as shown in FIG. 15D, thep-value 602 of 0.131 suggests that the null hypothesis of normality isnot inappropriately rejected.

Any specific manifestations of these and other similar example processesare not intended to be limiting to the invention. Any suitablemanifestation of these and other similar example processes can beselected within the scope of the illustrative embodiments.

Thus, a computer implemented method, system or apparatus, and computerprogram product are provided in the illustrative embodiments forcorrecting low-resolution measurements and other related features,functions, or operations. Where an embodiment or a portion thereof isdescribed with respect to a type of device, the computer implementedmethod, system or apparatus, the computer program product, or a portionthereof, are adapted or configured for use with a suitable andcomparable manifestation of that type of device.

Where an embodiment is described as implemented in an application, thedelivery of the application in a Software as a Service (SaaS) model iscontemplated within the scope of the illustrative embodiments. In a SaaSmodel, the capability of the application implementing an embodiment isprovided to a user by executing the application in a cloudinfrastructure. The user can access the application using a variety ofclient devices through a thin client interface such as a web browser, orother light-weight client-applications. The user does not manage orcontrol the underlying cloud infrastructure including the network,servers, operating systems, or the storage of the cloud infrastructure.In some cases, the user may not even manage or control the capabilitiesof the SaaS application. In some other cases, the SaaS implementation ofthe application may permit a possible exception of limited user-specificapplication configuration settings.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on a dedicated measurement system 124 or user's computer,partly on the user's computer or measurement system 124 as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server, etc. In thelatter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general-purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

All features disclosed in the specification, including the claims,abstract, and drawings, and all the steps in any method or processdisclosed, may be combined in any combination, except combinations whereat least some of such features and/or steps are mutually exclusive. Eachfeature disclosed in the specification, including the claims, abstract,and drawings, can be replaced by alternative features serving the same,equivalent, or similar purpose, unless expressly stated otherwise.

What is claimed is:
 1. A method comprising: receiving a plurality oflow-resolution measurements, the plurality of low-resolutionmeasurements corresponding to a plurality of unobservablehigh-resolution measurements; introducing variation in the plurality oflow-resolution measurements by iteratively computing, until atermination criteria is met, corresponding perturbed values for thelow-resolution measurements, said corresponding perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements; and running, responsive to the introducing, a distributiontest on final perturbed values that remain after said terminationcriteria is met.
 2. The method of claim 1, wherein the introducingcomprises: computing, for each low-resolution measurement, a firstinterval that contains a corresponding unobservable high-resolutionmeasurement corresponding to said each low-resolution measurement;generating, for each low-resolution measurement, a random observationfrom a uniform distribution on a defined interval; transforming eachrandom observation to be uniform on a second interval that correspondsto a distribution function of the first interval to obtain correspondingrescaled uniform observations, said distribution function being based onestimated distribution parameters of said low-resolution measurements;and inverse transforming, responsive to the transforming, and using aninverse of the distribution function, said rescaled uniform observationsto obtain said corresponding perturbed values, wherein the transformingand inverse transforming are repeated iteratively using new estimateddistribution parameters of the corresponding perturbed values until saidtermination criteria is met.
 3. The method of claim 2, wherein the firstinterval is computed based on a range of possible values of theunobservable high-resolution data.
 4. The method of claim 2, wherein thedistribution function is a cumulative distribution function.
 5. Themethod of claim 2, wherein the unobservable high-resolution measurementsare tested for normality and the estimated distribution parameters are amean and a standard deviation.
 6. The method of claim 5, wherein theestimated distribution parameters are a sample mean and a samplestandard deviation of the low-resolution measurements and the newestimated distribution parameters are a new sample mean and a new samplestandard deviation of the corresponding perturbed values.
 7. The methodof claim 2, wherein said corresponding perturbed values have a higherresolution than another resolution of the low-resolution measurements.8. The method of claim 1, wherein the low-resolution measurements arerounded versions of the unobservable high-resolution measurements. 9.The method of claim 1, wherein the termination criteria is a stablestandard deviation condition.
 10. The method of claim 1, wherein thedistribution test is an Anderson-Darling test.
 11. The method of claim10, wherein the Anderson-Darling test tests for normality.
 12. Themethod of claim 1, further comprising: obtaining the received set oflow-resolution measurements by measuring values of a defined property,using a low-resolution measurement device.
 13. The method of claim 12,wherein the values of a defined property are quantitative values.
 14. Anon-transitory computer readable storage medium storing programinstructions which, when executed by a processor, causes the processorto perform a procedure comprising the steps of: receiving a plurality oflow-resolution measurements, the plurality of low-resolutionmeasurements corresponding to a plurality of unobservablehigh-resolution measurements; introducing variation in the plurality oflow-resolution measurements by iteratively computing, until atermination criteria is met, corresponding perturbed values for thelow-resolution measurements, said corresponding perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements; and running, responsive to the introducing, a distributiontest on final perturbed values that remain after said terminationcriteria is met.
 15. The non-transitory computer readable storage mediumof claim 14, wherein the program instructions further cause theprocessor to perform the introducing by: computing, for eachlow-resolution measurement, a first interval that contains acorresponding unobservable high-resolution measurement corresponding tosaid each low-resolution measurement; generating, for eachlow-resolution measurement, a random observation from a uniformdistribution on a defined interval; transforming each random observationto be uniform on a second interval that corresponds to a distributionfunction of the first interval to obtain corresponding rescaled uniformobservations, said distribution function being based on estimateddistribution parameters of said low-resolution measurements; and inversetransforming, responsive to the transforming, and using an inverse ofthe distribution function, said rescaled uniform observations to obtaincorresponding perturbed values, wherein the transforming and inversetransforming are repeated iteratively using new estimated distributionparameters of the corresponding perturbed values until said terminationcriteria is met.
 16. The non-transitory computer readable storage mediumof claim 14, wherein the distribution test is an Anderson-Darling test.17. A computer system comprising: at least one processor configured toperforms the steps of: receiving a plurality of low-resolutionmeasurements, the plurality of low-resolution measurements correspondingto a plurality of unobservable high-resolution measurements; introducingvariation in the plurality of low-resolution measurements by iterativelycomputing, until a termination criteria is met, corresponding perturbedvalues for the low-resolution measurements, said corresponding perturbedvalues having a higher resolution than another resolution of thelow-resolution measurements; and running, responsive to the introducing,a distribution test on final perturbed values that remain after saidtermination criteria is met.
 18. The computer system of claim 17,wherein the at least one processor is configured to perform theintroducing step by: computing, for each low-resolution measurement, afirst interval that contains a corresponding unobservablehigh-resolution measurement corresponding to said each low-resolutionmeasurement; generating, for each low-resolution measurement, a randomobservation from a uniform distribution on a defined interval;transforming each random observation to be uniform on a second intervalthat corresponds to a distribution function of the first interval toobtain corresponding rescaled uniform observations, said distributionfunction being based on estimated distribution parameters of saidlow-resolution measurements; and inverse transforming, responsive to thetransforming, and using an inverse of the distribution function, saidrescaled uniform observations to obtain corresponding perturbed values,wherein the transforming and inverse transforming are repeatediteratively using new estimated distribution parameters of thecorresponding perturbed values until said termination criteria is met.